The electronic band structures of InN, GaN, and a hypothetical ordered ${\text{InGaN}}_{2}$ compound, all in the wurtzite crystal structure, are calculated using the quasiparticle self-consistent GW approximation. This approach leads to band gaps which are significantly improved compared to gaps calculated on the basis of the local approximation to density functional theory, although generally overestimated by 0.2\char21{}0.3 eV in comparison with experimental gap values. Details of the electronic energies and the effective masses including their pressure dependence are compared with available experimental information. The band gap of ${\text{InGaN}}_{2}$ is considerably smaller than what would be expected by linear interpolation implying a significant band gap bowing in InGaN alloys.